https://doi.org/10.3765/sp.10.8
Quantification over alternative intensions
*Thomas Ede Zimmermann Frankfurt
Submitted2016-08-15/Decision2016-11-02/Revision received2016-11-08/Decision 2016-11-27/Revision received2017-02-03/Accepted2017-02-05/Published2017 -06-16/Final typesetting2020-07-23
Abstract In footnote 13 on p. 85f. of his dissertation, Mats Rooth (1985)
addresses certain peculiarities of his treatment ofonly as a quantifier over propositions. The current note elaborates on that footnote to conclude that the lack of adequacy of this approach to quantification is more severe than previously thought. Section1presents a gap in the alternative[s] semantics
treatment of only. In Section 2 an attempt is made to close it by way of
meaning postulates to eliminate ‘degenerate’ models (Rooth’s term) in which extensions do not vary enough across Logical Space. In view of the lack of feasibility and systematicity of that approach, Section3 explores a more
principled, yet ultimately futile, strategy for determining ‘realistic’ models (Rooth’s term) that reflect the extensional variation offered by Model Space as a whole. Section4points out the limitations any such repair encounters
when it comes to sentences with non-contingent at-issue contents. Section5
briefly discusses a variant of the interpretation ofonly as a quantifier over propositional alternatives and how it fares with respect to the problems addressed in the previous sections.
Keywords: alternative semantics, possible worlds semantics, model-theoretic se-mantics, meaning postulates
* Some of the material in this note was presented in talks about semantic values and model-theoretic semantics delivered at McGill (Linguistics Department), the University of Amster-dam (DIP Colloquium), and the University of Konstanz (Zukunftskolleg) in2014. I am indebted to my audiences for numerous critical remarks as well as to David Beaver, Cécile Meier, Mats Rooth, and three anonymous referees for detailed comments on previous versions, which resulted in dramatic changes of the presentation. Thanks are also due to Carolina Cárdenas for helping me with typesetting the pre-final version of the manuscript.
Introduction
One of the principal assets to alternative semantics, originating withRooth 1985, is its straightforward account of the truth-conditional effects of certain
focus-sensitive elements likeonly, which does without any (possibly problem-atic) movement operations.1 At the same time it has long been known that
the somewhat underspecified analyses of alternative semantics do not always get the truth conditionsquiteright. The present note scrutinises the lacunae in the alternative treatment ofonly. More specifically, it addresses possible enhancements to alternative semantics by eliminating certain unintended models so as to predict more specific and adequate truth conditions for sentences withonly. Three objections will be raised against such attempts:
• In Section2, it will be argued that the standard technique of employing meaning postulates to make up for the lack of specificity in the truth conditions does not appear to be feasible.
• In Section3, a more principled approach to adequate truth conditions
in terms of a model-theoretic reflection technique, will be explored but ultimately dismissed since it turns out to be not viable either.
• In Section4, a specific counter-example will be presented to show that
the inadequacy of the truth conditions is sometimes beyond repair.
For expository reasons, these arguments will only be directed at the simple, uniform account of only as a propositional quantifier developed in Rooth 1985: ch. II, where in particular, quantification over alternatives to individual
referents is simulated by quantification over alternative propositions. Section
5will turn to an example of a more flexible (and more popular) approach to
alternative semantics and show that the latter leads to the same problems, albeit only in a sub-class of examples where only focus-associates with a proper part of its argument and quantifies over VP-denotations instead of propositions; as it turns out, the inadequacies of propositional quantifica-tion carry over to quantifiers over properties. The final secquantifica-tion then briefly discusses how these results may affect other analyses and theories beyond alternative treatments ofonly as a quantifier. Before all this, Section1 will
illustrate the core problem by going through a few pertinent examples.
1See, e.g.,Rooth1985:33ff.;Stechow1991:813f.;Kratzer1991:827f. for arguments adduced
1 Quantification over propositions vs. quantification over individuals
According to one version of alternative semantics, focus-sensitive operators like only quantify over propositions rather than individuals.2 As a case in
point, a sentence like:
(1) Only Mary is asleep.
comes out as expressing:3
(2) (∀p)[[∨p∧(∃x) p= ∧S(x)]→p=∧S(m)]
rather than the more straightforward predicate logic formalisation:
(3) (∀x)[S(x)→x=m]
Models abound in which(2)comes out true where it should not. Thus, e.g., if
John and Mary, while being distinct, happen to be asleep in exactly the same worlds (of a model) and are also the only ones who are asleep in the actual world (of that model), then (2) is true in the actual world (of that model) because the proposition denoted by ‘∧S(m)’ — that Mary is asleep — is the same proposition as the one expressed by ‘∧S(j)’ — that John is asleep — and it is the only proposition of the form ‘∧S(x)’ that is true in the actual world (of the model).4Intuitively, however,(1)should not count as true in that world (of
the model) because the extension ofS— the constant expressing the property
2Cf.Rooth 1985: ch. III, where a cross-categorial approach to focus-sensitive particles is developed. Other approaches to alternative quantification — like the one in Rooth1985: ch. II, which gets the truth conditions in constellations like(1)right — are plagued by the same problems when it comes to slightly more involved constellations but will not be addressed before Section5. Certain aspects of both approaches, such as context dependence, domain restrictions, compositionality, and multiple or wide foci, will be neglected in the following, since they are irrelevant for the comparison with ordinary quantification; only the conventional implicature triggered byonly will make a short appearance in Section4.
3Apart from the self-explanatory (and eliminable) terms for n-tuples of individuals, the
intensional type logic formulae used here are largely in line with Montague’s notation (Montague 1970). In particular, ‘∧’ abstracts over the (implicit) world parameter, and ‘∨’
indicates application to the world of evaluation; thus ifϕis a truth-valuable formula, ‘∧ϕ’
stands for the proposition expressed byϕ; and ifπ stands for a proposition, the formula ‘∨π’ says thatπ is true (of the world at stake).
of being asleep — contains two distinct individuals. And obviously(3)does
bring out this truth condition correctly.
More generally, asRooth(1985) (p.85f fn.13) observed, given the standard
modeling of propositions as regions in Logical Space, this analysis requires a certain abundance of possible worlds:(2)only comes out as equivalent to (3) if(4) holds, i.e., if the proposition denoted by ‘∧S(m)’ differs from the
proposition denoted by ‘∧S(x)’ for anyx other than Mary:
(4) (∀x)[x ≠m→∧S(x)
≠∧S(m)]
Generalising(4)from(1)by abstracting from Mary, the equivalence of (2)and (3) turns out to impose a certain degree of granularity on the propositions
expressed by simple predications:
(5) (∀x)(∀y)[x≠y→∧S(x)
≠∧S(y)]
The possible worlds setting as such does not guarantee the truth of(5). In
particular,(5)does not hold in models in which the numbernof individuals
by far exceeds the numbermof worlds in thatn>2m: for somex andy the propositions denoted by ‘∧S(x)’ and ‘∧S(y)’ would have to coincide, since there are only 2m propositions to begin with. Similarly, if n > 2 and the intension ofShappens to be rigid, the propositions denoted by ‘∧S(x)’ and ‘∧S(y)’ coincide for at least twox andy in the (constant) extension ofS. In order to guarantee(4), then, such ‘degenerate’ models (Rooth 1985) would have to be discarded by semantic theory. This could be achieved by way of meaning postulates (or similar constraints) to the effect that the lexical predicates must satisfy(5)in lieu ofS.5However, the above reasoning extends
way beyond the realm of lexical intransitives. Thus, e.g., in order for the interpretations(b) of the sentences(a)under (6)–(9)to come out right, the
corresponding assumptions(c)turn out to be crucial:6
(6) a. John only metMary.
b. (∀p)[[ ∨p∧(∃x) p=∧M(j, x)]→p=∧M(j,m)] c. (∀x1)(∀x2)(∀y1)(∀y2)[(x1, x2)≠(y1, y2)→
∧M(x
1, x2)≠∧M(y1, y2)]
(7) a. John only introducesMaryto Sue.
5SeeBeaver & Clark2008:84for an explicit suggestion along these lines.
b. (∀p)[[ ∨p∧(∃x) p=∧I(j,x,s)]∧ →p=∧I(j,m,s)]
c. (∀x1)(∀x2)(∀x3)(∀y1)(∀y2)(∀y3)[(x1, x2, x3)≠(y1, y2, y3)
→∧I(x
1, x2, x3)≠∧I(y1, y2, y3)]
(8) a. Only Mary is both drunk and asleep.
b. (∀p)[[ ∨p∧(∃x) p=∧[D(x)∧S(x)]]→p=∧[D(m)∧S(m)]] c. (∀x)(∀y)[x≠y →∧[D(x)∧S(x)]≠∧[D(y)∧S(y)]]
(9) a. John only knows thatMaryknows that Harry introduces Bill to
Sue.
b. (∀p)[[ ∨p∧(∃x) p=∧K(j,∧K(x,∧I(h,b,s)))]→
p=∧K(j,∧K(m,∧I(h,b,s)))] c. (∀x1) . . . (∀x5)(∀y1) . . . (∀y5)[(x1, . . . , x5)≠(y1, . . . , y5)→
∧K(x
1,∧K(x2,∧I(x3, x4, x5)))≠∧K(y1,∧K(y2,∧I(y3, y4, y5)))]
(6) and (7) illustrate that an adequate generalisation of (5) would have to
cover lexical verbs of higher valencies, like the binary and ternary predicates
M[eet] and I[ntroduce]. (8)and (9)indicate that the role of the predicateSin
(5)can be played by non-lexical (n-ary) predicates like:
(10) λx.[D(x)∧S(x)]
(11) λx5.λx4.λx3.λx2.λx1.K(x1,∧K(x2,∧I(x3, x4, x5)))
the general pattern being:7
(12) (∀→x )(- ∀→y )[- →x- ≠→y- →[∧R
---→
{x}]≠[∧R{---y→}]]
where ‘→x- ’ and ‘→y- ’ range overn-tuples of individuals (for fixed but arbitrary n).
2 Extensional Variation by Meaning Postulates
Closer inspection of the counter-examples to(5)reveals that there is
some-thing wrong with them for independent reasons: such ‘degenerate’ models according to which Logical Space is severely limited, ought be done away with anyway. What distinguishes such models from more ‘realistic’ (Rooth1985
p.85f fn.13) contenders, is their austerity: rather than varying freely across
Logical Space, they restrict the range of possible extensions ofS[leep] in that
it does not cover every set (of individuals). This defect could be remedied in a natural way by demanding a maximal degree of extensional variation:
(13) (∀X)♦[S=X]
where ‘X’ ranges over arbitrary sets of individuals. It is readily seen that
(13)implies (5) (though notvice versa).8 In the case of MandI, too, it may
seem natural to derive the pertinent requirements(6c)and(7c)from some
more general principle of variation, analogous to (13). Arguably, however,
in these cases the variation is conceptually bounded. In particular, it would seem that the extension of Mneeds to be irreflexive; for it does not make (literal) sense for anyone to meet him- or herself. Similarly, one may want to exclude from the possible extensions ofIany triples the third components of which coincide with either of the other two; for no one can introduce anyone (or anything) to him- or herself — on either reading of that clause.9 Further,
possibly more clear-cut cases along these lines may be found in comparatives and degree verbs likeexceed. Given these considerations, it appears dubious that (13) should be postulated for lexical predicates across the board. If anything, the range of possible extensions needs to be determined predicate by predicate. On top of(13)we may thus have:
(14) (∀R)[[¬(∃x)R(x, x)]→♦[M=R]]
(15) (∀S)[[¬(∃x)(∃y)[S(x, y, x)∨S(x, y, y)]]→♦[I=S]]
where ‘R’ and ‘S’ respectively range over binary and ternary relations, and so on, for any pertinent lexical predicate. However, this cannot be the whole story. For conjoined possibility does not imply compossibility, and so while
both D[runk] and S may have an unlimited range, (5) and itsD-variant do
8If(13)holds, then for any individualxthere is a world at which the extension ofSis the singleton{x}, thus excluding any individualy distinct fromx and satisfying(5). On the other hand if, e.g., the extension ofScould be any singleton but nothing else,(5)would hold, but(13)would not.(13)strengthens, and is in the spirit of the notion of,lexical freedom as defined inKeenan1987(methinks), which implies that for anyX, there is some lexical predicatePsatisfying ‘♦[P=X]’.
9To the extent that time travel into the past is a coherent concept, it might seem that persons
not exclude models according to which they are contradictories, say, and thus the extension of(10)is necessarily empty. To ensure(8c), the variation
postulates would thus have to cover the compound predicate in(10):
(16) (∀X) ♦[λx.[D(x)∧S(x)]=X]
Similarly, for(9c), one would need:
(17) (∀T )♦[[λx5.λx4.λx3.λx2.λx1.K(x1,∧K(x2,∧I(x3,x4,x5)))]=T]
with ‘T’ ranging over five-place relations among individuals.
Apart from idiosyncratic limitations due to lexical meanings as in(14)–(16), further restrictions on extensional variation ensue from interdependencies of lexical meanings. Thus, e.g., while unlimited variation seems to be desir-able for the intersection of logically independent predicates like D and S, constraints would have to be imposed on the possible extensions when it comes to Boolean combinations of, say,SandA[wake],KandB[elieve], etc. In fact, the range of extensional variation of complex predicates seems to be largely delimited by independent meaning postulates that the variation principles would have to somehow take into account — and it is not clear how this could be done in a systematic way.
3 Extensional variation by reflection principles
Formulating extensional variation postulates for ever more complex pre-dicates appears neither feasible nor particularly insightful. Thus a more principled way to proceed ought to be found; and, indeed, the examples of the previous section suggest a direction the search may take. For the kind of extensional variation that would secure the equivalence of(2)and(3)in a
over individuals in terms of quantification over properties: not all models are rich enough. We will then define what it means for a model’s Logical Space to be as rich as Model Space; such models will be said to reflect Model Space. The section ends with an argument that such reflective models are unlikely to exist.
Let us first see how the richness of Model Space affects the predicateS. If D and W are arbitrary domains of individuals and possible worlds, the following holds:
(18) For anyX ⊆D, there is aK0-modelM=(D, W ,FM)and a worldw ∈W
such that :
FM(S)(w)=X
whereK0is the class of all models of intensional type logic. While(13)grants
the predicate S a maximal amount of extensional variation across Logical Space,(18) records that the same kind of variation can be observed across Model Space. Yet the latter variation does not depend on(13):K0also contains
models that do not satisfy(13) at every world — which means that they do
not satisfy(13)at any world.10
In a similar vein, the extensions of M and I vary wildly across Model Space — even in the absence of meaning postulates(14)and (15):
(19) For any R ⊆ D2 there is a K0-model M = (D, W , FM) and a world
w ∈W such that :
FM(M)(w)=R.
For any S ⊆ D3 there is a K
0-model M = (D, W , FM) and a world
w ∈W such that :
FM(I)(w)=S.
As a consequence, the variation noted in(19)abides if Model Space is
restric-ted by irreflexivity postulates forMandI:
(20) a. ¬(∃x)M(x, x)
b. ¬(∃x)(∃y)[I(x, y, x)∨I(x, y, y)]
In other words, ifK1is the class of models satisfying (20), then (21)holds:
(21) a. For any irreflexiveR ⊆ D2 there is a K
1-model M = (D, W , FM)
and a worldw∈W such that :
FM(M)(w)=R.
b. For any irreflexive3 S ⊆ D3 there is a K1-model M = (D, W , FM)
and a worldw∈W such that :
FM(I)(w)=S.
where irreflexivity3 is the property attributed to (the extension of)Iin(20b).
Moreover, it is readily shown (not here, though) that the unlimited variation does not stop at compound predicates like(10)and(11), whose extensions vary freely across Model Space, independently of any trans-world variation principles like(16)and(17).
It is important to realise that cross-model variation cannot make up for a lack of cross-world variation.11Yet it is the latter that leads to a lack
of intensions in general and propositions in particular, which again may have dramatic consequences such as the non-equivalence of (2) and (3).
Since propositions, and intensions in general, are constructed by abstracting from worlds rather than models, the varying extensions of the latter are inaccessible to them. However, one may still hope that some, ‘realistic’ models would capture enough cross-model variation within their Logical Spaces. More concretely, given a classK of models, a modelM∗may be said toreflect K in that for anyM∈K and anyM-worldw there is anM∗-worldw∗in which the same (type-logical) sentences are true:
(22) {ϕ|Mîw ϕ} = {ϕ
M∗îw∗ϕ}12
In particular, a model that reflects a class of models across which the exten-sion of a (definable) predicate variesad libitum, makes any combination of attributions of that predicate true of at least one of its worlds. Hence, e.g., any setΣof sentences of the form ‘S(n)’, where ‘n’ is an individual constant, will have to be true at some world wΣ. And if pertinent postulates
guaran-tee the distinctness of sufficiently many constants, it would seem that the equivalence of(2)and(3)ought to emerge.
It is therefore natural to look for Rooth’s ‘realistic’ models among the ones that reflect the class K ∗ of models satisfying all pertinent meaning postulates, including irreflexivity assumptions like (20) but excluding the
11See alsoZimmermann2011:798f. for this point.
12‘îw’ relates (type-logical) modelsMto thesentences(closed, truth-valuable formulae) that
variation principles(13)–(17); after all, the intended effect of the latter shows
in the variation acrossK ∗and should thus be reflected in anyM∗satisfying
(22). Alas, this search strategy is futile. To see this, one may consider a
random contingency:
(23) Mary is asleep.
Clearly,K0contains modelsM0and M1according to which(23)expresses a
contradiction and a possibility, respectively:
(24) a. M06îw S(m), for anyM0-worldw;
b. M1îw0 S(m), for someM1-worldw0.
With the help of the unrestricted possibility operator ‘♦’, (24) may be
ex-pressed as a statement about arbitraryM0- and M1-worlds wand w0:13
(25) a. M0îw ¬♦S(m) b. M1îw0 ♦S(m)
In the (assumed) absence of any lexical postulates restricting the extensional range of the constants ‘S’ and ‘m’, K ∗, too, ought to contain modelsM
0
and M1 satisfying (24) and (25). Hence a ‘realistic’ model M∗ that reflects
K ∗would have to contain worldsw
0andw1 that reflect arbitraryM0- and
M1-worldsw andw0:
(26) a. M∗îw0¬♦S(m)
b. M∗6î
w1♦S(m)
But the formulae in (26) are modally closed and so the truth of neither
depends on the particular choice of worldsw0andw1; hence(27)would have
to hold for allw∗in M∗’s Logical Space:
(27) M∗î
w∗ [[¬♦S(m)]∧♦S(m)]
. . . which cannot be.14 So the strategy of narrowing down the class of all
models to the more realistic ones in one fell swoop, leads to a contradiction.
13Again, the two formulae under scrutiny are modally closed. Without loss of generality, one
may actually assume thatM0andM1share their Logical Space and hence thatw=w0; this
is so because Model Space is closed under arbitrary isomorphisms.
14The situation is vaguely reminiscent of modal logic: in the canonical model of the logic of universal accessibility (S5), accessibility is an equivalence relation but not universal; cf.
One may try and escape this conclusion by excluding modal formulae like the ones in (25) from the reflection requirement (22). Again, it is not
obvious how this could be done in a sensible way. In particular, the above reasoning does not depend on the fact that the formulae under scrutiny are modally closed; a conjunction of either with some contingent sentence would have done just as well. Moreover, while it is crucial that the formulae involve intensionality, i.e., abstraction from particular worlds, restricting(22)
to purely extensionalϕ would make the reflection property unattractively weak. Thus, e.g., in order for (9a) and its focus variants to come out right,
a ‘realistic’ model ought to reflect the extensional range of the property in
(11)and thus make arbitrary combinations of attribution of it true, as long as
they obey the relevant postulates (like the veridicality ofK). However, these attributions are not extensional in that they involve the propositional attitude K[nowledge].
4 Limits of propositional quantification
Since it does not seem to be possible to formulate general principles of ex-tensional variation that would us allow to derive them, the relevant instances of(12)— here repeated as(28)— would have to be assumed as meaning
pos-tulates in their own right:
(28) (∀→x )(- ∀→y )[- →x- 6=→y- →[∧R{→x- }]6=[∧R{→y- }]] [=(12)]
The problem is to define these instances. To begin with, (28) cannot be assumed for arbitrary (n-place) relationsR; for its universal closure(29)turns
out to be inconsistent:
(29) (∀R)(∀→x )(- ∀→y )[- →x- 6=y→- →[∧R{→x- }]6=[∧R{→y- }]]
Among others, certain trivial properties such as self-identity, or being ident-ical with some fixedn-tuple, contradict(29). Hence(29)ought to be suitably
relativised by some (2nd order) property℘, if only for consistence:
(30) (∀R)[℘(R)→(∀→x )(- ∀→y )[- →x- 6=→y- →∧R{→x- } 6=∧R{→y- }]]
It is not clear how℘should be specified so as to imply all relevant instances of (28), including (5) and (6c)–(9c). In particular, filtering out those R that
contradict (28) will not do:(28) is satisfied by precisely the relations that
Hence in(30),℘cannot be the property℘∗of being consistent with(28): given
any relationR, either℘∗(R) holds or its negation does; andRis inconsistent with (28) precisely in the latter case. So the properties that are consistent
with(28)are the ones that satisfy(28), which means that equating℘with℘∗
would turn (30)into the tautology that anyRthat satisfies(28), satisfies(28).
The problem, then, is to find a property℘shared by relations likeS, L, I,(10),
and(11)so that (28)would make sure that they also have ℘∗.
One may thus ask what the relations mentioned have in common. For one thing, they are all definable in terms of the lexical predicates featuring in indirect interpretation.15However,℘must not be identified with definability
in S, L, I, K, etc.; for not all definable relationsR can be assumed to satisfy
(28): as already indicated, at least some of them, like self-identity (which is definable in these constants, even without them), should be excluded for logical reasons.16
In order to define a suitable restriction ℘, one may try to characterise those relations that are definable from the logical representations of sen-tences like(1)and(6a)–(9a). In principle, this could be done in terms of an
indirect interpretation algorithm. Yet again, this method of defining℘would overshoot unless at least tautologies and contradictions are exempted from it. In fact, MatsRooth(1985p.85fn.13) already observed that the strategy of mimicking individual by propositional quantification reaches its limits when it comes to rigid intensions:17
(31) Only three is an odd number.
(31)is clearly false, and so is a straightforward predicate logic formalisation of it:
(32) (∀x)[O(x)→x=3]
However, (31) comes out true on an analysis of only as a quantifier over
propositions:
(33) (∀p)[[∨p∧(∃x)p=∧O(x)]→p=∧O(3)]
15Byindirect interpretationI mean a framework along the lines ofMontague1970where, instead of directly giving their semantic values (as in, say,Heim & Kratzer1998), an algorithm is specified that takes (the LFs of) natural language expressions to their type-logical translations.
16Actually, self-identity does satisfy(28)in totally ‘degenerate’ models with just one individual. 17For expository reasons,Rooth(1985) uses a slightly more involved example to prove the
The reason for this failure lies in a peculiarity of the predicate O[dd num-ber], whose extension is taken to not vary across Logical Space.18 Hence, in
particular, any proposition of the form ‘∧O(x)’ will coincide with either the empty set or all of Logical Space. So the latter is the only true proposition of that form, thus verifying (33). Unlike in the case of (1) and (2), though,
there is no escape from the conclusion that (33) is an inadequate
formal-isation of the truth conditions of (31), which, again, seem aptly captured
by(32). One may therefore doubt that the failure to adequately capture the
truth-conditions of (31) should be adduced to the notorious lack of
fine-grainedness of possible world semantics, as Rooth(p. 85f fn. 13) seems to
suggest: after all, the adequate formalisation(32), too, can be expressed in
that very framework. Moreover, the inadequacies due to rigid intensions may lead to contingencies:19
(34) Only Mary is one of John and Mary and exactly as tall as either one.
Clearly, under the (hardly objectionable) assumption thatJohnandMary are (rigid) names of different persons and the predicatebe one of John and Mary rigidly denotes the pair of them,(34)ought to come out as a contradiction.
And again it does if the sentence is analysed in terms of quantification over individuals:
(35) (∀x)[[[x =j∨x=m]∧(∀y)[[y =j∨y =m]→h(x)=h(y)]]→
x=m]
To see what (35) comes down to, one should first notice that its matrix (=
the scope of the outermost universal quantifier) is trivially satisfied by any individual distinct from John: Mary satisfies it in view of the triviality of the consequent (of the matrix); and any other individual satisfies it because it trivially fails to satisfy the left conjunct of the antecedent. As a consequence, the universal quantification in (35) boils down to:
18As in the case of proper names (cf. fn.4above), this would have to be guaranteed by a specific meaning postulate, e.g., ‘(∃X)O=X’.
19Slightly less artificial examples may be obtained if the domain of quantification is kept fixed:
Only this object is exactly as heavy as everything else.— The rest of this section deviates from the Early Access version, which contained a faulty predicate logic analysis of(34)instead of
[[[j=j∨j=m]∧(∀y)[[y =j∨y =m]→h(j)=h(y)]]→j=m]
which contains a redundant conjunct and reduces to:
[(∀y)[[y =j∨y =m]→h(j)=h(y)]→j=m]
which in turn is equivalent, by the distinctness assumption, to the negation of its antecedent and thus (by familiar quantifier laws) to:
(∃y)[[y =j∨y =m]∧h(j)≠h(y)]
Eliminating the existential quantifier yields:
[h(j)≠h(j)∨h(j)≠h(m)]
which boils down to:
(36) h(j)≠h(m)
This is not yet the desired contradiction, which only arises once the conven-tional implicature triggered byonly is taken into consideration. Before we get to it, though, let us take a look at the ordinary content that alternative semantics ascribes to(34):
(37) (∀p)[[∨p∧(∃x)p= ∧[[x=j∨x=m]∧(∀y)[[y =j∨y =m]
→h(x)=h(y)]]]
→p= ∧[[m=j∨m=m]∧(∀y)[[y =j∨y =m]→h(m)=h(y)]]]
Closer inspection reveals that (37) is logically valid. To see this, one may
observe that(37)can be equivalently rewritten as the tautology(38), where
pj≈m abbreviates ‘∧[h(j)=h(m)]’:
(38) (∀p)[[∨p∧p=pj≈m]→p=pj≈m]
The details of the reformulation are kindly left to the reader. As in the case of
(31), then, the alternative treatment misanalyses a contradiction,(34), as valid.
readability.20 In the case of sentences containing only, it can be identified
with the (ordinary) content of the sentence withoutonly. As a case in point, the predicate logic and alternative formalisations of the full conventional content of(1)come out as (40)and(39), respectively, where the second line
gives a logically equivalent (‘≡’) reformulation of the first:
(39) [(∀x)[S(x)→x=m]∧S(m)]
≡(∀x)[S(x)↔x=m]
(40) [(∀p)[[∨p∧(∃x)p=∧S(x)]→p=∧S(m)]∧S(m)]
≡[(∀p)[[∨p∧(∃x)p=∧S(x)]↔p=∧S(m)]]
By the same token, the full content of(31)ought to be (41), as predicted by
standard predicate logic formalisation, but comes out as(42)in alternative
semantics:
(41) (∀x)[O(x)↔x=3]
(42) (∀p)[[∨p∧(∃x) p=∧O(x)]↔p=∧O(3)]
Since the ordinary content (32) of (31) is already contradictory, so is the
stronger full content(41). On the other hand,(42)only adds the uninformative
conjunct that3is odd to the trivial ordinary content(33)of(31), and is thus also trivially true. So on both analyses of (31), ordinary content and full
content coincide. This may be taken to confirm the suspicion that (31) is
an exceptional or even neurotic case, somewhat beyond the proper area of application of possible worlds semantics.
However, things stand differently with(34). Given that, according to the
alternative analysis, its ordinary content comes out as trivial, its full content coincides with its conventional implicature, i.e., the proposition that Mary is [one of John and Mary and] as tall as [either Mary or] John:
(43) [(∀p)[[∨p∧p=pj≈m]→p=pj≈m]∧
[[m=j∨m=m]∧(∀y)[[y =j∨y =m]→h(m)=h(y)]]]
≡[(∀p)[[∨p∧p=pj≈m]→p=pj≈m]∧h(m)=h(j)]
≡h(m)=h(j)
20The analysis in Rooth1985: 120ff. takes both assertions and implicatures into account.
On the predicate logic analysis, on the other hand, the conventional im-plicature of(34)is(44), which does, obviously, contradict its ordinary content (36), as expected and desired:
(44) [[m=j∨m=m]∧(∀y)[[y =j∨y =m]→h(m)=h(j)]]
≡h(m)=h(j)
(34)differs from run-of-the-mill examples for lacking fine-grainedness: though
some apparent contingencies are known to come out as contradictions in pos-sible worlds semantics21, contradictions that express contingent propositions
are unheard of (to my knowledge, anyway). The fact that certain contradict-ory sentences are misanalysed as expressing contingent propositions should not be blamed on the notorious lack of fine-grainedness of the framework alone: after all, the predicate logic account ofonly does get(34)right and is
within the reach of possible worlds semantics. Hence the example may and should be taken as indication against the analysis ofonly as a propositional quantifier in possible worlds semantics.
5 Quantification over alternative properties
The upshot of the above reasoning is that quantification over individuals cannot be fully captured by quantification over propositions. But then the latter seems to be a peculiarity of the cross-categorial treatment ofRooth 1985: ch. III anyway: later analyses in the tradition of alternative semantics
followRooth1985: ch. II andRooth1992in treatingonlyas quantifying over
alternative intensions of the constituent it modifies. It may thus seem that the above considerations do not bear on these approaches where (alternat-ive) propositions give way to (alternat(alternat-ive) intensions in general. Indeed,(1)
then does come out as directly quantifying over alternatives to proper name denotations — and thus as a quantifier over individuals, just as the predicate logic formalisation (3) would have it.22 However, if only modifies a larger
constituent that (properly) contains the focussed name it associates with, the alternative intensions of this constituent — which need not be propositions (and typically are not) — are plagued by the very same potential lack of
vari-21Cf.Kaplan1995andKripke2011for interesting cases.
22These denotations may be referents or intensions, depending on the details of the framework.
ation deplored in connection with propositions above, and thus susceptible to the same kind of reasoning.
To begin with, the argument of the previous section may be adapted to certain cases in whichonly modifies a proposition-denoting clause:
(45) Bill knows only thatMaryis one of John and Mary and exactly as tall
as either one.
It appears that(45)is contradictory in that its full content implies(46), which
can only be true if Bill believes that both Mary and John satisfy(47)(in lieu of
x), thus contradicting(45):
(46) Bill knows that Maryis one of John and Mary and exactly as tall as
either one.
(47) Bill knows only thatxis one of John and Mary and exactly as tall as
either one.
These examples are admittedly marginal and hardly the stuff that kills a theory. However, when quantification over individuals is mimicked by quan-tification over intensions other than propositions, the worries addressed in Sections1–3return in a different guise. As a case in point, instead of analysing
(48a)in terms of alternatives to its conventional implicature, as in(6b)above
(and repeated below), alternatives to the predicate intensions, as in (48c), are
supposed to achieve the intended effect:23
(48) a. John only metMary. [=(6a)]
b. (∀p)[[ ∨p∧(∃y) p= ∧M(j, y)]→p= ∧M(j,m)] [≈(
6b)]
c. (∀P )[[ P{j} ∧(∃y) P = xˆM(x, y)]→P = xˆM(x,m)]
where ‘P’ ranges over properties, i.e., possible intensions of unary predicates. One may wonder whether (48c)fares better than (48b)with respect to the problems discussed above. To begin with, it should be noted that the two formulae are not equivalent. For although(48c)implies(48b), this implication
does not reverse.24
23‘ ˆx ϕ’ abbreviates ‘∧λx. ϕ’, thus denoting the intension of aλ-abstracted predicate.
24To show this, one may consider a (totally ‘degenerate’) model with one world w and
four individuals h,j,m, and s that assigns to M the characteristic function of the set
Despite an increase in granularity, the analysis(48c) still cannot escape
the worries and objections previously raised against quantification over propositions. In fact, a closer look at counter-models to “(48b) ⇒ (48c)”
mentioned in the preceding paragraph (and fn.24) reveals that properties
are almost as indiscriminate as propositions: though(48a) does come out
false (as it should), the analogous(49a)comes out true (though it shouldn’t)
if analysed as(49b):
(49) a. Harry only metSue.
b. (∀P )[[ P{h} ∧(∃y) P = xˆM(x, y)]→P = xˆM(x,s)]
The reason for this again lies in a lack of fine-grainedness: ‘ ˆxM(x,j)’ and ‘ ˆxM(x,s)’ denote the same property, whose only possible extension is{h}, and this is the only property of the form ‘ ˆx M(x, y)’ that h has (at w), just as (49b) requires. So (49b) is true even though, according to the only
world of the model described, the referent ofHarry stands in the relation denoted byMto more than one individual. Needless to say, even for the above ‘degenerate’ model, a predicate logic formalisation takingonly as quantifying over individuals would get the truth values of both(48a)and (49a)right.
The example illustrates that the more general approach involving predic-ate intensions inherits the deficiencies of propositional quantification. This may come as a surprise in view of the adequacy of the predicate logic form-alisation, according to whichonlyis a quantifier over individuals and thus ought to be less discriminative than a quantifier over alternative properties. However, the finesse of the latter is lost on the individuals to be quantified over as the sample analysis(48c)indicates and, more generally, a survey of the three quantifiers in question makes clear:
(50) a. λy. λQ. (∀z) [Q{z} →z = y]
b. λp. λA. (∀q) [[∨q∧A(q)]→q = p]
c. λx. λP . λS. (∀S) [[S{x} ∧S(S)]→S = P ]
The quantifier in (50a)applies to (the referent of) the focussed elementy0
associated withonly and (the intension of) the predicateQ0expressed by the
rest of the sentence (minusonly); the latter may be obtained by a standardλ -abstraction mechanism.25Taking(48a)as an example,y
0would be the bearer
of the nameMary, andQ0 the property denoted by ‘ˆzM(j, z)’. Thus(50a)is
a special case of a folklore analysis of only as a binary (non-conservative) quantifier over individuals.26
The quantifier in (50b), on the other hand, applies to the proposition
p0 = {w|y0∈Q0(w)} expressed by the whole sentence (minusonly) and
the setA of its alternatives (including p0itself):
A = {q|(∃z) q= {w|z∈Q0(w)}}
In the above example (48a), p0 would thus be the proposition denoted by
‘∧M(j,m)’, and A would be the set of propositions of the form ‘∧M(j, z)’. Hence(50b)is Rooth’spropositional quantifier analysisRooth1985: 120, with
some minor (and mostly notational) changes.27
The quantifier in (50c) applies to (the referent of) the subject x0, (the
intension of) the (clause) predicateP0, and the setS of its alternatives:
S = {S|(∃z)(∀w)S(w)=R0(w)(z)}
where R0 is constructed from the predicate (minus the focussed element)
by a λ-abstraction device, in analogy to the construction of Q0 from the
whole sentence; in particular, for any world w: P0(w)=R0(w)(y0). In the
case of(48a),R0would coincide with the relation expressed byM. And the
three arguments would be: the referent of the nameJohn(=x0), the property
denoted by ‘ ˆxM(x,m)’ (=P0), and the set of properties of the form ‘ ˆxM(x,z)’
(= S).(50c)is adapted from the ‘domain selection’ analysis of Rooth1985:
44, according to whichonly may quantify over alternative VP-denotations.28
If analysed as the quantifier in(50a),only imposes the assumedly correct
truth condition on the resulting sentence, viz., that the singleton{y0}cover
the extension ofQ0. On the other hand, a careful comparison of the above
26Cf.Geach1980[1962]:207ff. for an early source.
27In particular, Rooth’s Russellian format has been Frege-Churched (in the sense ofKaplan 1975); the variable ranging over the set of alternatives has beenλ-bound (and renamed from ‘C’ to‘A’); and the conventional implicature has been omitted, in line with the analyses in Section1.
28This ‘direct’ account ofonly as a quantifier over properties, which can be found in most
alternative semantics approaches fromRooth1992onward, must not be confused with the type-shifted version derived from(50b)in the cross-categorial account ofRooth1985: ch. III, which reads (on p.121, again glossing over matters of notation and type regimentation):
(*) λx. λP . λA. (∀p)[[∨p∧ A(p)]→p = ∧P{x}]
quantifiers brings out the inadequacy of both(50b)and(50c).29On the one
hand, the observations in Section1already indicate why the adequate truth
conditions are not guaranteed to be expressible in terms of the two arguments of the quantifier in (50b). And despite its more complex type and finer
granularity, the quantifier in(50c)does not attain the adequacy of the truth
conditions imposed by (50a) either. Loosely speaking, the reason is that
its extra argument is wasted on the subject instead of relating directly to the focussed element: the injective (one-one) function required for the equivalence between(50c) and(50a)does not concern the subject argument.
Of course, this does not make the quantifier in (50c) as such inadequate,
but only the way it is matched with its linguistic environment, following the alternative semantics approach. In fact, if applied to the focussed element
y0, the propertyQ0of abstracting from it, and the set of all properties that
nothing buty0has, the quantifier(50c)would impose the truth conditions
of the full content; but then this procedure would not be in line with the alternative semantics architecture.
6 Conclusion
Now the we have seen that not all is well with the alternative treatment of only as a quantifier over propositions,30 we may wonder whether similar
deficiencies could not be found in other applications of alternative semantics, or in other theoretical frameworks that make use of similar assumptions and analyses. Given the wide-ranging literature on these topics, I will have to confine myself to some general remarks here.
The cause for all the trouble observed above is that the truth conditions of certain sentences involve quantification over individuals, which is mimicked by quantification over propositions. In terms of (two-sorted) types this means that operators on the domain(et)are ‘coded’ by corresponding operators on the domain ((st)t). The mismatch between these domains has been observed before and elsewhere, notably in the semantics of interrogatives, where it has also been held responsible for a number of inadequacies of
29Rigorous proofs can be found on p.15ff. of the archived first draft of this paper: http: //semanticsarchive.net/Archive/mRhOGNjN/propquant.pdf.
proposition-based approaches.31 More specifically (and ignoring irrelevant
intensional parameters), the alternative analysis essentially replaces the (apparently adequate) quantifierOonly of type((et)(et)), defined in(51), by an operatorO−of type(((st)t)((st)t)), satisfying the equivalence in(
52)(for
all individualsx and propertiesP):
(51) λE.λx. (∀y)[E(y)→y =x] [≈(50a)]
(52) O−(∧[P{x}], λp.(∃y) p= ∧[P{y}]) ≡ Oonly(x,∨P )
The discussion in Sections 1–4 brought out that the equivalence (52) does
not hold for all instances and that its exact scope is not easy to define. Since similar assumptions have been made concerning the treatment of other focus-sensitive elements within alternative semantics, these arguments are likely to carry over to them. As a case in point, according to a standard analysis ofeven, it contributes a (non-at-issue) proposition by operating on the alternative intensions of its scope. Again, the situation is essentially as in (51) and (52), with (53) being the operator Oeven (of type ((s(et))(et))) that expresses the intended reading, and(54)the equivalence condition that
alternative semantics imposes on its surrogate (of type(((st)t)((st)t))):32
(53) λP .λx. (∀y)µ(P{x})≤µ(P{y})
(54) O−(∧[P{x}], λp.(∃y) p= ∧[P{y}]) ≡ Oeven(x, P )
It should not come as a surprise that(54)leads to the very same problems
as the alternative reduction(52)ofonly; finding the relevant examples and
adapting the above arguments is left to the reader. A more principled invest-igation into the domain switching strategy underlying alternative semantics will have to be left to future research on type-shifting.
So is the alternative semantics analysis ofonly doomed to assign inad-equate truth conditions? Maybe not. Part of the trouble is the identification of propositions with sets of possible worlds. As Mats Rooth (op. cit.) already
31See, e.g.,Groenendijk & Stokhof1984, p.280ff.,Zimmermann1985, p.436f.,Krifka2001, orAloni et al.2007for pertinent remarks and observations concerning leading frameworks of interrogative semantics. A reviewer also pointed out an interesting connection between question semantics and the discussion in Sections1–3: ‘The danger would be that in degen-erate models,MaryandBillwould incorrectly count as the same answer to the questionWho left?.’
pointed out, going ‘ “more intensional” ’ — along the lines of property the-ory33— might be an option. However, rather than replacing the indeterminacy
of intended models by an indeterminacy of fine-grained properties, one may stick to the possible worlds framework but incorporate (parts of) a competing semantic approach to focus: the theory of structured propositions and mean-ings,34 which offers precisely the degree of granularity needed to capture
the truth conditions of ordinary quantification over individuals. As a case in point,(1)could be accounted for adequately after replacing propositions in (2)by pairs of properties and individuals, as in (55):
(1) Only Mary is asleep
(2) (∀p)[[∨p∧(∃x) p= ∧S(x)]→p=∧S(m)]
(55) (∀π )[[↓π∧(∃x) π =(S, x)]→π=(S,m)]
whereπ ranges over structured meanings (of appropriate types) and↓ evalu-ates them in a way that ‘↓(P , x)’ comes down to ‘P{x}’. It is not hard to see that(55)is equivalent to the intended analysis(3):35
(3) (∀x)[S(x)→x=m]
In any case it would seem that life in the alternative paradise that Rooth has created does not come for free.
33Bealer1982. Rooth quotesChierchia1984in this connection.
34I am indebted to Irene Heim (p.c., June2014) for bringing up this suggestion. SeeStechow1991 andKrifka2001for expositions and comparison of the frameworks. In fact, the structured alternatives employed byOnea2013: ch.8come close to this kind of hybrid architecture.
35This may be established by the following chain ofIL-equivalences:
(∀π )[[↓π∧(∃x) π= (S, x)]→π =(S,m)]
≡ (∀π )[(∃Q)(∃y)π=(Q, y)→[[↓π∧(∃x) π = (S, x)]→π =(S,m)]]
def. of struct. prop.
≡ (∀π )(∀Q)(∀y)[π =(Q, y)→[[↓π∧(∃x) π= (S, x)]→π =(S,m)]]
quantifier law
≡ (∀Q)(∀y)[[↓(Q, y)∧(∃x) (Q, y)=(S, x)]→(Q, y)=(S,m)] identity law
≡ (∀Q)(∀y)[[Q{y} ∧(∃x) (Q, y)=(S, x)]→(Q, y)=(S,m)] def. of↓ ≡ (∀Q)(∀y)[[Q{y} ∧(∃x)[ Q=S∧x=y]]→[Q=S∧y=m)]] def. of pair
≡ (∀Q)(∀y)[Q=S→[Q{y} →[Q=S∧y =m]]] predicate logic
≡ (∀y)[S(y)→[S=S∧y=m]] identity law + prop. logic
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Thomas Ede Zimmermann Institut für Linguistik Goethe-Universität Norbert-Wollheim-Platz1 60629Frankfurt
Germany
